Properties

 Label 2304.c1 Conductor $2304$ Discriminant $13824$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

 $$y^2=x^3-6x$$ y^2=x^3-6x (homogenize, simplify) $$y^2z=x^3-6xz^2$$ y^2z=x^3-6xz^2 (dehomogenize, simplify) $$y^2=x^3-6x$$ y^2=x^3-6x (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -6, 0])

gp: E = ellinit([0, 0, 0, -6, 0])

magma: E := EllipticCurve([0, 0, 0, -6, 0]);

oscar: E = EllipticCurve([0, 0, 0, -6, 0])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

magma: MordellWeilGroup(E);

Infinite order Mordell-Weil generator and height

 $P$ = $$\left(3, 3\right)$$ (3, 3) $\hat{h}(P)$ ≈ $0.84419079932496752692050478991$

sage: E.gens()

magma: Generators(E);

gp: E.gen

Torsion generators

$$\left(0, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

Integral points

$$(-2,\pm 2)$$, $$\left(0, 0\right)$$, $$(3,\pm 3)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

 Conductor: $$2304$$ = $2^{8} \cdot 3^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $13824$ = $2^{9} \cdot 3^{3}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$1728$$ = $2^{6} \cdot 3^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.51601946832452311334053968298\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.3105329259115095182522750833\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $\dots$ Szpiro ratio: $2.1942649571203137\dots$

BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.84419079932496752692050478991\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.3506911865035923203380772326\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $4$  = $2\cdot2$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $2.8286226710255914456288696422$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

BSD formula

$\displaystyle 2.828622671 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.350691 \cdot 0.844191 \cdot 4}{2^2} \approx 2.828622671$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

Modular invariants

$$q - 2 q^{5} - 4 q^{13} + 8 q^{17} + O(q^{20})$$

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 128
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $2$ $III$ Additive 1 8 9 0
$3$ $2$ $III$ Additive 1 2 3 0

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 2304.c consists of 2 curves linked by isogenies of degree 2.

Twists

This elliptic curve is its own minimal quadratic twist.

The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $216$.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.55296.2 $$\Z/4\Z$$ Not in database $8$ 8.4.12230590464.5 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.12230590464.4 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.36691771392.3 $$\Z/6\Z$$ Not in database $8$ 8.0.1528823808000.10 $$\Z/10\Z$$ Not in database $16$ 16.0.149587343098087735296.14 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ 16.0.2393397489569403764736.45 $$\Z/8\Z$$ Not in database $16$ 16.0.1346286087882789617664.1 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/10\Z$$ Not in database $16$ 16.4.5385144351531158470656.13 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ord ss ss ord ord ss ss ord ss ord ord ss ss - - 1 1,1 1,3 1 3 1,1 1,1 1 1,1 1 1 1,1 1,1 - - 0 0,0 0,0 0 0 0,0 0,0 0 0,0 0 0 0,0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.